Source code for networkx.algorithms.planar_drawing

from collections import defaultdict

import networkx as nx

__all__ = ["combinatorial_embedding_to_pos"]

[docs] def combinatorial_embedding_to_pos(embedding, fully_triangulate=False): """Assigns every node a (x, y) position based on the given embedding The algorithm iteratively inserts nodes of the input graph in a certain order and rearranges previously inserted nodes so that the planar drawing stays valid. This is done efficiently by only maintaining relative positions during the node placements and calculating the absolute positions at the end. For more information see [1]_. Parameters ---------- embedding : nx.PlanarEmbedding This defines the order of the edges fully_triangulate : bool If set to True the algorithm adds edges to a copy of the input embedding and makes it chordal. Returns ------- pos : dict Maps each node to a tuple that defines the (x, y) position References ---------- .. [1] M. Chrobak and T.H. Payne: A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989 """ if len(embedding.nodes()) < 4: # Position the node in any triangle default_positions = [(0, 0), (2, 0), (1, 1)] pos = {} for i, v in enumerate(embedding.nodes()): pos[v] = default_positions[i] return pos embedding, outer_face = triangulate_embedding(embedding, fully_triangulate) # The following dicts map a node to another node # If a node is not in the key set it means that the node is not yet in G_k # If a node maps to None then the corresponding subtree does not exist left_t_child = {} right_t_child = {} # The following dicts map a node to an integer delta_x = {} y_coordinate = {} node_list = get_canonical_ordering(embedding, outer_face) # 1. Phase: Compute relative positions # Initialization v1, v2, v3 = node_list[0][0], node_list[1][0], node_list[2][0] delta_x[v1] = 0 y_coordinate[v1] = 0 right_t_child[v1] = v3 left_t_child[v1] = None delta_x[v2] = 1 y_coordinate[v2] = 0 right_t_child[v2] = None left_t_child[v2] = None delta_x[v3] = 1 y_coordinate[v3] = 1 right_t_child[v3] = v2 left_t_child[v3] = None for k in range(3, len(node_list)): vk, contour_nbrs = node_list[k] wp = contour_nbrs[0] wp1 = contour_nbrs[1] wq = contour_nbrs[-1] wq1 = contour_nbrs[-2] adds_mult_tri = len(contour_nbrs) > 2 # Stretch gaps: delta_x[wp1] += 1 delta_x[wq] += 1 delta_x_wp_wq = sum(delta_x[x] for x in contour_nbrs[1:]) # Adjust offsets delta_x[vk] = (-y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2 y_coordinate[vk] = (y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2 delta_x[wq] = delta_x_wp_wq - delta_x[vk] if adds_mult_tri: delta_x[wp1] -= delta_x[vk] # Install v_k: right_t_child[wp] = vk right_t_child[vk] = wq if adds_mult_tri: left_t_child[vk] = wp1 right_t_child[wq1] = None else: left_t_child[vk] = None # 2. Phase: Set absolute positions pos = {} pos[v1] = (0, y_coordinate[v1]) remaining_nodes = [v1] while remaining_nodes: parent_node = remaining_nodes.pop() # Calculate position for left child set_position( parent_node, left_t_child, remaining_nodes, delta_x, y_coordinate, pos ) # Calculate position for right child set_position( parent_node, right_t_child, remaining_nodes, delta_x, y_coordinate, pos ) return pos
def set_position(parent, tree, remaining_nodes, delta_x, y_coordinate, pos): """Helper method to calculate the absolute position of nodes.""" child = tree[parent] parent_node_x = pos[parent][0] if child is not None: # Calculate pos of child child_x = parent_node_x + delta_x[child] pos[child] = (child_x, y_coordinate[child]) # Remember to calculate pos of its children remaining_nodes.append(child) def get_canonical_ordering(embedding, outer_face): """Returns a canonical ordering of the nodes The canonical ordering of nodes (v1, ..., vn) must fulfill the following conditions: (See Lemma 1 in [2]_) - For the subgraph G_k of the input graph induced by v1, ..., vk it holds: - 2-connected - internally triangulated - the edge (v1, v2) is part of the outer face - For a node v(k+1) the following holds: - The node v(k+1) is part of the outer face of G_k - It has at least two neighbors in G_k - All neighbors of v(k+1) in G_k lie consecutively on the outer face of G_k (excluding the edge (v1, v2)). The algorithm used here starts with G_n (containing all nodes). It first selects the nodes v1 and v2. And then tries to find the order of the other nodes by checking which node can be removed in order to fulfill the conditions mentioned above. This is done by calculating the number of chords of nodes on the outer face. For more information see [1]_. Parameters ---------- embedding : nx.PlanarEmbedding The embedding must be triangulated outer_face : list The nodes on the outer face of the graph Returns ------- ordering : list A list of tuples `(vk, wp_wq)`. Here `vk` is the node at this position in the canonical ordering. The element `wp_wq` is a list of nodes that make up the outer face of G_k. References ---------- .. [1] Steven Chaplick. Canonical Orders of Planar Graphs and (some of) Their Applications 2015 .. [2] M. Chrobak and T.H. Payne: A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989 """ v1 = outer_face[0] v2 = outer_face[1] chords = defaultdict(int) # Maps nodes to the number of their chords marked_nodes = set() ready_to_pick = set(outer_face) # Initialize outer_face_ccw_nbr (do not include v1 -> v2) outer_face_ccw_nbr = {} prev_nbr = v2 for idx in range(2, len(outer_face)): outer_face_ccw_nbr[prev_nbr] = outer_face[idx] prev_nbr = outer_face[idx] outer_face_ccw_nbr[prev_nbr] = v1 # Initialize outer_face_cw_nbr (do not include v2 -> v1) outer_face_cw_nbr = {} prev_nbr = v1 for idx in range(len(outer_face) - 1, 0, -1): outer_face_cw_nbr[prev_nbr] = outer_face[idx] prev_nbr = outer_face[idx] def is_outer_face_nbr(x, y): if x not in outer_face_ccw_nbr: return outer_face_cw_nbr[x] == y if x not in outer_face_cw_nbr: return outer_face_ccw_nbr[x] == y return outer_face_ccw_nbr[x] == y or outer_face_cw_nbr[x] == y def is_on_outer_face(x): return x not in marked_nodes and (x in outer_face_ccw_nbr or x == v1) # Initialize number of chords for v in outer_face: for nbr in embedding.neighbors_cw_order(v): if is_on_outer_face(nbr) and not is_outer_face_nbr(v, nbr): chords[v] += 1 ready_to_pick.discard(v) # Initialize canonical_ordering canonical_ordering = [None] * len(embedding.nodes()) canonical_ordering[0] = (v1, []) canonical_ordering[1] = (v2, []) ready_to_pick.discard(v1) ready_to_pick.discard(v2) for k in range(len(embedding.nodes()) - 1, 1, -1): # 1. Pick v from ready_to_pick v = ready_to_pick.pop() marked_nodes.add(v) # v has exactly two neighbors on the outer face (wp and wq) wp = None wq = None # Iterate over neighbors of v to find wp and wq nbr_iterator = iter(embedding.neighbors_cw_order(v)) while True: nbr = next(nbr_iterator) if nbr in marked_nodes: # Only consider nodes that are not yet removed continue if is_on_outer_face(nbr): # nbr is either wp or wq if nbr == v1: wp = v1 elif nbr == v2: wq = v2 else: if outer_face_cw_nbr[nbr] == v: # nbr is wp wp = nbr else: # nbr is wq wq = nbr if wp is not None and wq is not None: # We don't need to iterate any further break # Obtain new nodes on outer face (neighbors of v from wp to wq) wp_wq = [wp] nbr = wp while nbr != wq: # Get next neighbor (clockwise on the outer face) next_nbr = embedding[v][nbr]["ccw"] wp_wq.append(next_nbr) # Update outer face outer_face_cw_nbr[nbr] = next_nbr outer_face_ccw_nbr[next_nbr] = nbr # Move to next neighbor of v nbr = next_nbr if len(wp_wq) == 2: # There was a chord between wp and wq, decrease number of chords chords[wp] -= 1 if chords[wp] == 0: ready_to_pick.add(wp) chords[wq] -= 1 if chords[wq] == 0: ready_to_pick.add(wq) else: # Update all chords involving w_(p+1) to w_(q-1) new_face_nodes = set(wp_wq[1:-1]) for w in new_face_nodes: # If we do not find a chord for w later we can pick it next ready_to_pick.add(w) for nbr in embedding.neighbors_cw_order(w): if is_on_outer_face(nbr) and not is_outer_face_nbr(w, nbr): # There is a chord involving w chords[w] += 1 ready_to_pick.discard(w) if nbr not in new_face_nodes: # Also increase chord for the neighbor # We only iterator over new_face_nodes chords[nbr] += 1 ready_to_pick.discard(nbr) # Set the canonical ordering node and the list of contour neighbors canonical_ordering[k] = (v, wp_wq) return canonical_ordering def triangulate_face(embedding, v1, v2): """Triangulates the face given by half edge (v, w) Parameters ---------- embedding : nx.PlanarEmbedding v1 : node The half-edge (v1, v2) belongs to the face that gets triangulated v2 : node """ _, v3 = embedding.next_face_half_edge(v1, v2) _, v4 = embedding.next_face_half_edge(v2, v3) if v1 in (v2, v3): # The component has less than 3 nodes return while v1 != v4: # Add edge if not already present on other side if embedding.has_edge(v1, v3): # Cannot triangulate at this position v1, v2, v3 = v2, v3, v4 else: # Add edge for triangulation embedding.add_half_edge(v1, v3, ccw=v2) embedding.add_half_edge(v3, v1, cw=v2) v1, v2, v3 = v1, v3, v4 # Get next node _, v4 = embedding.next_face_half_edge(v2, v3) def triangulate_embedding(embedding, fully_triangulate=True): """Triangulates the embedding. Traverses faces of the embedding and adds edges to a copy of the embedding to triangulate it. The method also ensures that the resulting graph is 2-connected by adding edges if the same vertex is contained twice on a path around a face. Parameters ---------- embedding : nx.PlanarEmbedding The input graph must contain at least 3 nodes. fully_triangulate : bool If set to False the face with the most nodes is chooses as outer face. This outer face does not get triangulated. Returns ------- (embedding, outer_face) : (nx.PlanarEmbedding, list) tuple The element `embedding` is a new embedding containing all edges from the input embedding and the additional edges to triangulate the graph. The element `outer_face` is a list of nodes that lie on the outer face. If the graph is fully triangulated these are three arbitrary connected nodes. """ if len(embedding.nodes) <= 1: return embedding, list(embedding.nodes) embedding = nx.PlanarEmbedding(embedding) # Get a list with a node for each connected component component_nodes = [next(iter(x)) for x in nx.connected_components(embedding)] # 1. Make graph a single component (add edge between components) for i in range(len(component_nodes) - 1): v1 = component_nodes[i] v2 = component_nodes[i + 1] embedding.connect_components(v1, v2) # 2. Calculate faces, ensure 2-connectedness and determine outer face outer_face = [] # A face with the most number of nodes face_list = [] edges_visited = set() # Used to keep track of already visited faces for v in embedding.nodes(): for w in embedding.neighbors_cw_order(v): new_face = make_bi_connected(embedding, v, w, edges_visited) if new_face: # Found a new face face_list.append(new_face) if len(new_face) > len(outer_face): # The face is a candidate to be the outer face outer_face = new_face # 3. Triangulate (internal) faces for face in face_list: if face is not outer_face or fully_triangulate: # Triangulate this face triangulate_face(embedding, face[0], face[1]) if fully_triangulate: v1 = outer_face[0] v2 = outer_face[1] v3 = embedding[v2][v1]["ccw"] outer_face = [v1, v2, v3] return embedding, outer_face def make_bi_connected(embedding, starting_node, outgoing_node, edges_counted): """Triangulate a face and make it 2-connected This method also adds all edges on the face to `edges_counted`. Parameters ---------- embedding: nx.PlanarEmbedding The embedding that defines the faces starting_node : node A node on the face outgoing_node : node A node such that the half edge (starting_node, outgoing_node) belongs to the face edges_counted: set Set of all half-edges that belong to a face that have been visited Returns ------- face_nodes: list A list of all nodes at the border of this face """ # Check if the face has already been calculated if (starting_node, outgoing_node) in edges_counted: # This face was already counted return [] edges_counted.add((starting_node, outgoing_node)) # Add all edges to edges_counted which have this face to their left v1 = starting_node v2 = outgoing_node face_list = [starting_node] # List of nodes around the face face_set = set(face_list) # Set for faster queries _, v3 = embedding.next_face_half_edge(v1, v2) # Move the nodes v1, v2, v3 around the face: while v2 != starting_node or v3 != outgoing_node: if v1 == v2: raise nx.NetworkXException("Invalid half-edge") # cycle is not completed yet if v2 in face_set: # v2 encountered twice: Add edge to ensure 2-connectedness embedding.add_half_edge(v1, v3, ccw=v2) embedding.add_half_edge(v3, v1, cw=v2) edges_counted.add((v2, v3)) edges_counted.add((v3, v1)) v2 = v1 else: face_set.add(v2) face_list.append(v2) # set next edge v1 = v2 v2, v3 = embedding.next_face_half_edge(v2, v3) # remember that this edge has been counted edges_counted.add((v1, v2)) return face_list